On the structure of the Gram matrix for Gabor systems generated by B-splines

We consider the Gabor system $\mathcal{G}(g,a\mathbb{Z}\times b\mathbb{Z})$ generated by a continuous, compactly supported function $g$ over the time-frequency lattice generated by the parameters $a$ and $b$. We show that, under an appropriate ordering of the Gabor elements, certain submatrices of the Gram matrix of $\mathcal{G}(g,a\mathbb{Z}\times b\mathbb{Z})$ exhibit a block-Toeplitz structure. This structural property enables us to derive spectral results for finite sub-blocks of the Gram matrix by appealing to the spectral theory of Toeplitz matrices. In particular, we apply our results to the Gram matrix of Gabor systems generated by the $N$th-order B-spline.